Mastery Check Solutions

Show What You Know

A polynomial function never contains breaks or sharp points. Therefore the graph is always smooth and continuous.

Sketch an example and counterexample of a polynomial function.

Sample example

Sample counterexample

Note

Does your sketch represent an even or odd function? Is the leading coefficient positive or negative?

The set of coefficients for a polynomial function $a_{n}, a_{n - 1}, . . ., a_{2}, a_{1}, a_{0}$ are sometimes rational numbers.

Explain your response to part C.

Polynomial functions have coefficients that are real numbers. Real numbers include the subsets of rational or irrational numbers. Rational numbers also include the subsets of integers, whole numbers, and natural numbers.

Note

Your student can review the real number diagram if they do not recall the subsets of real numbers.

The end behavior of polynomial function is always determined by the leading coefficient test.

Explain your response to part E.

The leading coefficient test allows you to generalize the end behavior of a function using the leading coefficient a and the degree n.

Determine if the degree of the function is even or odd and if the leading coefficient is positive or negative. Then generalize the end behavior.

Sketch 1

Sketch 2

Sketch 3

Sketch 4

even
a = positive $x \to \pm \infty, f (x) \to + \infty$

odd
a = positive $x \to - \infty, f (x) \to - \infty,$ and $x \to + \infty, f (x) \to + \infty$

odd
a = negative $x \to - \infty, f (x) \to + \infty,$ and $x \to + \infty, f (x) \to - \infty$

even
a = negative $x \to \pm \infty, f (x) \to - \infty$

Note

While it may seem repetitive to write out the end behavior, it is critical to developing an understanding of how polynomial functions, their leading coefficient, and degree all work together.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Determine the type of function (polynomial, power, both, neither) represented by equations or graphs.
Describe the end behavior using a graph or equation, including the domain and range.